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<div class="title">IDRS.h</div>  </div>
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<div class="fragment"><div class="line"><a name="l00001"></a><span class="lineno">    1</span>&#160;<span class="comment">// This file is part of Eigen, a lightweight C++ template library</span></div>
<div class="line"><a name="l00002"></a><span class="lineno">    2</span>&#160;<span class="comment">// for linear algebra.</span></div>
<div class="line"><a name="l00003"></a><span class="lineno">    3</span>&#160;<span class="comment">//</span></div>
<div class="line"><a name="l00004"></a><span class="lineno">    4</span>&#160;<span class="comment">// Copyright (C) 2020 Chris Schoutrop &lt;c.e.m.schoutrop@tue.nl&gt;</span></div>
<div class="line"><a name="l00005"></a><span class="lineno">    5</span>&#160;<span class="comment">// Copyright (C) 2020 Jens Wehner &lt;j.wehner@esciencecenter.nl&gt;</span></div>
<div class="line"><a name="l00006"></a><span class="lineno">    6</span>&#160;<span class="comment">// Copyright (C) 2020 Jan van Dijk &lt;j.v.dijk@tue.nl&gt;</span></div>
<div class="line"><a name="l00007"></a><span class="lineno">    7</span>&#160;<span class="comment">//</span></div>
<div class="line"><a name="l00008"></a><span class="lineno">    8</span>&#160;<span class="comment">// This Source Code Form is subject to the terms of the Mozilla</span></div>
<div class="line"><a name="l00009"></a><span class="lineno">    9</span>&#160;<span class="comment">// Public License v. 2.0. If a copy of the MPL was not distributed</span></div>
<div class="line"><a name="l00010"></a><span class="lineno">   10</span>&#160;<span class="comment">// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.</span></div>
<div class="line"><a name="l00011"></a><span class="lineno">   11</span>&#160; </div>
<div class="line"><a name="l00012"></a><span class="lineno">   12</span>&#160;<span class="preprocessor">#ifndef EIGEN_IDRS_H</span></div>
<div class="line"><a name="l00013"></a><span class="lineno">   13</span>&#160;<span class="preprocessor">#define EIGEN_IDRS_H</span></div>
<div class="line"><a name="l00014"></a><span class="lineno">   14</span>&#160; </div>
<div class="line"><a name="l00015"></a><span class="lineno">   15</span>&#160;<span class="preprocessor">#include &quot;./InternalHeaderCheck.h&quot;</span></div>
<div class="line"><a name="l00016"></a><span class="lineno">   16</span>&#160; </div>
<div class="line"><a name="l00017"></a><span class="lineno">   17</span>&#160;<span class="keyword">namespace </span><a class="code" href="namespaceEigen.html">Eigen</a> {</div>
<div class="line"><a name="l00018"></a><span class="lineno">   18</span>&#160; </div>
<div class="line"><a name="l00019"></a><span class="lineno">   19</span>&#160;<span class="keyword">namespace </span>internal {</div>
<div class="line"><a name="l00034"></a><span class="lineno">   34</span>&#160;<span class="keyword">template</span> &lt;<span class="keyword">typename</span> Vector, <span class="keyword">typename</span> RealScalar&gt;</div>
<div class="line"><a name="l00035"></a><span class="lineno">   35</span>&#160;<span class="keyword">typename</span> <a class="codeRef" href="../classEigen_1_1DenseBase.html#a5feed465b3a8e60c47e73ecce83e39a2">Vector::Scalar</a> omega(<span class="keyword">const</span> <a class="codeRef" href="../group__matrixtypedefs.html#ga2623c0d4641dda067fbdb9a009ef0c91">Vector</a>&amp; t, <span class="keyword">const</span> <a class="codeRef" href="../group__matrixtypedefs.html#ga2623c0d4641dda067fbdb9a009ef0c91">Vector</a>&amp; s, RealScalar angle) {</div>
<div class="line"><a name="l00036"></a><span class="lineno">   36</span>&#160;  <span class="keyword">using</span> numext::abs;</div>
<div class="line"><a name="l00037"></a><span class="lineno">   37</span>&#160;  <span class="keyword">typedef</span> <span class="keyword">typename</span> <a class="codeRef" href="../classEigen_1_1DenseBase.html#a5feed465b3a8e60c47e73ecce83e39a2">Vector::Scalar</a> Scalar;</div>
<div class="line"><a name="l00038"></a><span class="lineno">   38</span>&#160;  <span class="keyword">const</span> RealScalar ns = s.stableNorm();</div>
<div class="line"><a name="l00039"></a><span class="lineno">   39</span>&#160;  <span class="keyword">const</span> RealScalar nt = t.stableNorm();</div>
<div class="line"><a name="l00040"></a><span class="lineno">   40</span>&#160;  <span class="keyword">const</span> Scalar ts = t.dot(s);</div>
<div class="line"><a name="l00041"></a><span class="lineno">   41</span>&#160;  <span class="keyword">const</span> RealScalar rho = <a class="codeRef" href="../namespaceEigen.html#ae27242789e7e62a8c42579b79be59b1a">abs</a>(ts / (nt * ns));</div>
<div class="line"><a name="l00042"></a><span class="lineno">   42</span>&#160; </div>
<div class="line"><a name="l00043"></a><span class="lineno">   43</span>&#160;  <span class="keywordflow">if</span> (rho &lt; angle) {</div>
<div class="line"><a name="l00044"></a><span class="lineno">   44</span>&#160;    <span class="keywordflow">if</span> (ts == Scalar(0)) {</div>
<div class="line"><a name="l00045"></a><span class="lineno">   45</span>&#160;      <span class="keywordflow">return</span> Scalar(0);</div>
<div class="line"><a name="l00046"></a><span class="lineno">   46</span>&#160;    }</div>
<div class="line"><a name="l00047"></a><span class="lineno">   47</span>&#160;    <span class="comment">// Original relation for om is given by</span></div>
<div class="line"><a name="l00048"></a><span class="lineno">   48</span>&#160;    <span class="comment">// om = om * angle / rho;</span></div>
<div class="line"><a name="l00049"></a><span class="lineno">   49</span>&#160;    <span class="comment">// To alleviate potential (near) division by zero this can be rewritten as</span></div>
<div class="line"><a name="l00050"></a><span class="lineno">   50</span>&#160;    <span class="comment">// om = angle * (ns / nt) * (ts / abs(ts)) = angle * (ns / nt) * sgn(ts)</span></div>
<div class="line"><a name="l00051"></a><span class="lineno">   51</span>&#160;    <span class="keywordflow">return</span> angle * (ns / nt) * (ts / <a class="codeRef" href="../namespaceEigen.html#ae27242789e7e62a8c42579b79be59b1a">abs</a>(ts));</div>
<div class="line"><a name="l00052"></a><span class="lineno">   52</span>&#160;  }</div>
<div class="line"><a name="l00053"></a><span class="lineno">   53</span>&#160;  <span class="keywordflow">return</span> ts / (nt * nt);</div>
<div class="line"><a name="l00054"></a><span class="lineno">   54</span>&#160;}</div>
<div class="line"><a name="l00055"></a><span class="lineno">   55</span>&#160; </div>
<div class="line"><a name="l00056"></a><span class="lineno">   56</span>&#160;<span class="keyword">template</span> &lt;<span class="keyword">typename</span> MatrixType, <span class="keyword">typename</span> Rhs, <span class="keyword">typename</span> Dest, <span class="keyword">typename</span> Preconditioner&gt;</div>
<div class="line"><a name="l00057"></a><span class="lineno">   57</span>&#160;<span class="keywordtype">bool</span> idrs(<span class="keyword">const</span> MatrixType&amp; A, <span class="keyword">const</span> Rhs&amp; b, Dest&amp; x, <span class="keyword">const</span> Preconditioner&amp; precond, <a class="codeRef" href="../namespaceEigen.html#a62e77e0933482dafde8fe197d9a2cfde">Index</a>&amp; iter,</div>
<div class="line"><a name="l00058"></a><span class="lineno">   58</span>&#160;          <span class="keyword">typename</span> Dest::RealScalar&amp; relres, <a class="codeRef" href="../namespaceEigen.html#a62e77e0933482dafde8fe197d9a2cfde">Index</a> S, <span class="keywordtype">bool</span> smoothing, <span class="keyword">typename</span> Dest::RealScalar angle,</div>
<div class="line"><a name="l00059"></a><span class="lineno">   59</span>&#160;          <span class="keywordtype">bool</span> replacement) {</div>
<div class="line"><a name="l00060"></a><span class="lineno">   60</span>&#160;  <span class="keyword">typedef</span> <span class="keyword">typename</span> Dest::RealScalar RealScalar;</div>
<div class="line"><a name="l00061"></a><span class="lineno">   61</span>&#160;  <span class="keyword">typedef</span> <span class="keyword">typename</span> Dest::Scalar Scalar;</div>
<div class="line"><a name="l00062"></a><span class="lineno">   62</span>&#160;  <span class="keyword">typedef</span> <a class="codeRef" href="../classEigen_1_1Matrix.html">Matrix&lt;Scalar, Dynamic, 1&gt;</a> VectorType;</div>
<div class="line"><a name="l00063"></a><span class="lineno">   63</span>&#160;  <span class="keyword">typedef</span> Matrix&lt;Scalar, Dynamic, Dynamic, ColMajor&gt; DenseMatrixType;</div>
<div class="line"><a name="l00064"></a><span class="lineno">   64</span>&#160;  <span class="keyword">const</span> <a class="codeRef" href="../namespaceEigen.html#a62e77e0933482dafde8fe197d9a2cfde">Index</a> N = b.size();</div>
<div class="line"><a name="l00065"></a><span class="lineno">   65</span>&#160;  S = S &lt; x.rows() ? S : x.rows();</div>
<div class="line"><a name="l00066"></a><span class="lineno">   66</span>&#160;  <span class="keyword">const</span> RealScalar tol = relres;</div>
<div class="line"><a name="l00067"></a><span class="lineno">   67</span>&#160;  <span class="keyword">const</span> <a class="codeRef" href="../namespaceEigen.html#a62e77e0933482dafde8fe197d9a2cfde">Index</a> maxit = iter;</div>
<div class="line"><a name="l00068"></a><span class="lineno">   68</span>&#160; </div>
<div class="line"><a name="l00069"></a><span class="lineno">   69</span>&#160;  <a class="codeRef" href="../namespaceEigen.html#a62e77e0933482dafde8fe197d9a2cfde">Index</a> replacements = 0;</div>
<div class="line"><a name="l00070"></a><span class="lineno">   70</span>&#160;  <span class="keywordtype">bool</span> trueres = <span class="keyword">false</span>;</div>
<div class="line"><a name="l00071"></a><span class="lineno">   71</span>&#160; </div>
<div class="line"><a name="l00072"></a><span class="lineno">   72</span>&#160;  FullPivLU&lt;DenseMatrixType&gt; lu_solver;</div>
<div class="line"><a name="l00073"></a><span class="lineno">   73</span>&#160; </div>
<div class="line"><a name="l00074"></a><span class="lineno">   74</span>&#160;  DenseMatrixType P;</div>
<div class="line"><a name="l00075"></a><span class="lineno">   75</span>&#160;  {</div>
<div class="line"><a name="l00076"></a><span class="lineno">   76</span>&#160;    HouseholderQR&lt;DenseMatrixType&gt; qr(DenseMatrixType::Random(N, S));</div>
<div class="line"><a name="l00077"></a><span class="lineno">   77</span>&#160;    P = (qr.householderQ() * DenseMatrixType::Identity(N, S));</div>
<div class="line"><a name="l00078"></a><span class="lineno">   78</span>&#160;  }</div>
<div class="line"><a name="l00079"></a><span class="lineno">   79</span>&#160; </div>
<div class="line"><a name="l00080"></a><span class="lineno">   80</span>&#160;  <span class="keyword">const</span> RealScalar normb = b.stableNorm();</div>
<div class="line"><a name="l00081"></a><span class="lineno">   81</span>&#160; </div>
<div class="line"><a name="l00082"></a><span class="lineno">   82</span>&#160;  <span class="keywordflow">if</span> (internal::isApprox(normb, RealScalar(0))) {</div>
<div class="line"><a name="l00083"></a><span class="lineno">   83</span>&#160;    <span class="comment">// Solution is the zero vector</span></div>
<div class="line"><a name="l00084"></a><span class="lineno">   84</span>&#160;    x.setZero();</div>
<div class="line"><a name="l00085"></a><span class="lineno">   85</span>&#160;    iter = 0;</div>
<div class="line"><a name="l00086"></a><span class="lineno">   86</span>&#160;    relres = 0;</div>
<div class="line"><a name="l00087"></a><span class="lineno">   87</span>&#160;    <span class="keywordflow">return</span> <span class="keyword">true</span>;</div>
<div class="line"><a name="l00088"></a><span class="lineno">   88</span>&#160;  }</div>
<div class="line"><a name="l00089"></a><span class="lineno">   89</span>&#160;  <span class="comment">// from http://homepage.tudelft.nl/1w5b5/IDRS/manual.pdf</span></div>
<div class="line"><a name="l00090"></a><span class="lineno">   90</span>&#160;  <span class="comment">// A peak in the residual is considered dangerously high if‖ri‖/‖b‖&gt; C(tol/epsilon).</span></div>
<div class="line"><a name="l00091"></a><span class="lineno">   91</span>&#160;  <span class="comment">// With epsilon the</span></div>
<div class="line"><a name="l00092"></a><span class="lineno">   92</span>&#160;  <span class="comment">// relative machine precision. The factor tol/epsilon corresponds to the size of a</span></div>
<div class="line"><a name="l00093"></a><span class="lineno">   93</span>&#160;  <span class="comment">// finite precision number that is so large that the absolute round-off error in</span></div>
<div class="line"><a name="l00094"></a><span class="lineno">   94</span>&#160;  <span class="comment">// this number, when propagated through the process, makes it impossible to</span></div>
<div class="line"><a name="l00095"></a><span class="lineno">   95</span>&#160;  <span class="comment">// achieve the required accuracy.The factor C accounts for the accumulation of</span></div>
<div class="line"><a name="l00096"></a><span class="lineno">   96</span>&#160;  <span class="comment">// round-off errors. This parameter has beenset to 10−3.</span></div>
<div class="line"><a name="l00097"></a><span class="lineno">   97</span>&#160;  <span class="comment">// mp is epsilon/C</span></div>
<div class="line"><a name="l00098"></a><span class="lineno">   98</span>&#160;  <span class="comment">// 10^3 * eps is very conservative, so normally no residual replacements will take place.</span></div>
<div class="line"><a name="l00099"></a><span class="lineno">   99</span>&#160;  <span class="comment">// It only happens if things go very wrong. Too many restarts may ruin the convergence.</span></div>
<div class="line"><a name="l00100"></a><span class="lineno">  100</span>&#160;  <span class="keyword">const</span> RealScalar mp = RealScalar(1e3) * NumTraits&lt;Scalar&gt;::epsilon();</div>
<div class="line"><a name="l00101"></a><span class="lineno">  101</span>&#160; </div>
<div class="line"><a name="l00102"></a><span class="lineno">  102</span>&#160;  <span class="comment">// Compute initial residual</span></div>
<div class="line"><a name="l00103"></a><span class="lineno">  103</span>&#160;  <span class="keyword">const</span> RealScalar tolb = tol * normb;  <span class="comment">// Relative tolerance</span></div>
<div class="line"><a name="l00104"></a><span class="lineno">  104</span>&#160;  VectorType r = b - A * x;</div>
<div class="line"><a name="l00105"></a><span class="lineno">  105</span>&#160; </div>
<div class="line"><a name="l00106"></a><span class="lineno">  106</span>&#160;  VectorType x_s, r_s;</div>
<div class="line"><a name="l00107"></a><span class="lineno">  107</span>&#160; </div>
<div class="line"><a name="l00108"></a><span class="lineno">  108</span>&#160;  <span class="keywordflow">if</span> (smoothing) {</div>
<div class="line"><a name="l00109"></a><span class="lineno">  109</span>&#160;    x_s = x;</div>
<div class="line"><a name="l00110"></a><span class="lineno">  110</span>&#160;    r_s = r;</div>
<div class="line"><a name="l00111"></a><span class="lineno">  111</span>&#160;  }</div>
<div class="line"><a name="l00112"></a><span class="lineno">  112</span>&#160; </div>
<div class="line"><a name="l00113"></a><span class="lineno">  113</span>&#160;  RealScalar normr = r.stableNorm();</div>
<div class="line"><a name="l00114"></a><span class="lineno">  114</span>&#160; </div>
<div class="line"><a name="l00115"></a><span class="lineno">  115</span>&#160;  <span class="keywordflow">if</span> (normr &lt;= tolb) {</div>
<div class="line"><a name="l00116"></a><span class="lineno">  116</span>&#160;    <span class="comment">// Initial guess is a good enough solution</span></div>
<div class="line"><a name="l00117"></a><span class="lineno">  117</span>&#160;    iter = 0;</div>
<div class="line"><a name="l00118"></a><span class="lineno">  118</span>&#160;    relres = normr / normb;</div>
<div class="line"><a name="l00119"></a><span class="lineno">  119</span>&#160;    <span class="keywordflow">return</span> <span class="keyword">true</span>;</div>
<div class="line"><a name="l00120"></a><span class="lineno">  120</span>&#160;  }</div>
<div class="line"><a name="l00121"></a><span class="lineno">  121</span>&#160; </div>
<div class="line"><a name="l00122"></a><span class="lineno">  122</span>&#160;  DenseMatrixType G = DenseMatrixType::Zero(N, S);</div>
<div class="line"><a name="l00123"></a><span class="lineno">  123</span>&#160;  DenseMatrixType U = DenseMatrixType::Zero(N, S);</div>
<div class="line"><a name="l00124"></a><span class="lineno">  124</span>&#160;  DenseMatrixType M = DenseMatrixType::Identity(S, S);</div>
<div class="line"><a name="l00125"></a><span class="lineno">  125</span>&#160;  VectorType t(N), v(N);</div>
<div class="line"><a name="l00126"></a><span class="lineno">  126</span>&#160;  Scalar om = 1.;</div>
<div class="line"><a name="l00127"></a><span class="lineno">  127</span>&#160; </div>
<div class="line"><a name="l00128"></a><span class="lineno">  128</span>&#160;  <span class="comment">// Main iteration loop, guild G-spaces:</span></div>
<div class="line"><a name="l00129"></a><span class="lineno">  129</span>&#160;  iter = 0;</div>
<div class="line"><a name="l00130"></a><span class="lineno">  130</span>&#160; </div>
<div class="line"><a name="l00131"></a><span class="lineno">  131</span>&#160;  <span class="keywordflow">while</span> (normr &gt; tolb &amp;&amp; iter &lt; maxit) {</div>
<div class="line"><a name="l00132"></a><span class="lineno">  132</span>&#160;    <span class="comment">// New right hand size for small system:</span></div>
<div class="line"><a name="l00133"></a><span class="lineno">  133</span>&#160;    VectorType f = (r.adjoint() * P).adjoint();</div>
<div class="line"><a name="l00134"></a><span class="lineno">  134</span>&#160; </div>
<div class="line"><a name="l00135"></a><span class="lineno">  135</span>&#160;    <span class="keywordflow">for</span> (<a class="codeRef" href="../namespaceEigen.html#a62e77e0933482dafde8fe197d9a2cfde">Index</a> k = 0; k &lt; S; ++k) {</div>
<div class="line"><a name="l00136"></a><span class="lineno">  136</span>&#160;      <span class="comment">// Solve small system and make v orthogonal to P:</span></div>
<div class="line"><a name="l00137"></a><span class="lineno">  137</span>&#160;      <span class="comment">// c = M(k:s,k:s)\f(k:s);</span></div>
<div class="line"><a name="l00138"></a><span class="lineno">  138</span>&#160;      lu_solver.compute(M.block(k, k, S - k, S - k));</div>
<div class="line"><a name="l00139"></a><span class="lineno">  139</span>&#160;      VectorType c = lu_solver.solve(f.segment(k, S - k));</div>
<div class="line"><a name="l00140"></a><span class="lineno">  140</span>&#160;      <span class="comment">// v = r - G(:,k:s)*c;</span></div>
<div class="line"><a name="l00141"></a><span class="lineno">  141</span>&#160;      v = r - G.rightCols(S - k) * c;</div>
<div class="line"><a name="l00142"></a><span class="lineno">  142</span>&#160;      <span class="comment">// Preconditioning</span></div>
<div class="line"><a name="l00143"></a><span class="lineno">  143</span>&#160;      v = precond.solve(v);</div>
<div class="line"><a name="l00144"></a><span class="lineno">  144</span>&#160; </div>
<div class="line"><a name="l00145"></a><span class="lineno">  145</span>&#160;      <span class="comment">// Compute new U(:,k) and G(:,k), G(:,k) is in space G_j</span></div>
<div class="line"><a name="l00146"></a><span class="lineno">  146</span>&#160;      U.col(k) = U.rightCols(S - k) * c + om * v;</div>
<div class="line"><a name="l00147"></a><span class="lineno">  147</span>&#160;      G.col(k) = A * U.col(k);</div>
<div class="line"><a name="l00148"></a><span class="lineno">  148</span>&#160; </div>
<div class="line"><a name="l00149"></a><span class="lineno">  149</span>&#160;      <span class="comment">// Bi-Orthogonalise the new basis vectors:</span></div>
<div class="line"><a name="l00150"></a><span class="lineno">  150</span>&#160;      <span class="keywordflow">for</span> (<a class="codeRef" href="../namespaceEigen.html#a62e77e0933482dafde8fe197d9a2cfde">Index</a> i = 0; i &lt; k - 1; ++i) {</div>
<div class="line"><a name="l00151"></a><span class="lineno">  151</span>&#160;        <span class="comment">// alpha =  ( P(:,i)&#39;*G(:,k) )/M(i,i);</span></div>
<div class="line"><a name="l00152"></a><span class="lineno">  152</span>&#160;        Scalar alpha = P.col(i).dot(G.col(k)) / M(i, i);</div>
<div class="line"><a name="l00153"></a><span class="lineno">  153</span>&#160;        G.col(k) = G.col(k) - alpha * G.col(i);</div>
<div class="line"><a name="l00154"></a><span class="lineno">  154</span>&#160;        U.col(k) = U.col(k) - alpha * U.col(i);</div>
<div class="line"><a name="l00155"></a><span class="lineno">  155</span>&#160;      }</div>
<div class="line"><a name="l00156"></a><span class="lineno">  156</span>&#160; </div>
<div class="line"><a name="l00157"></a><span class="lineno">  157</span>&#160;      <span class="comment">// New column of M = P&#39;*G  (first k-1 entries are zero)</span></div>
<div class="line"><a name="l00158"></a><span class="lineno">  158</span>&#160;      <span class="comment">// M(k:s,k) = (G(:,k)&#39;*P(:,k:s))&#39;;</span></div>
<div class="line"><a name="l00159"></a><span class="lineno">  159</span>&#160;      M.block(k, k, S - k, 1) = (G.col(k).adjoint() * P.rightCols(S - k)).adjoint();</div>
<div class="line"><a name="l00160"></a><span class="lineno">  160</span>&#160; </div>
<div class="line"><a name="l00161"></a><span class="lineno">  161</span>&#160;      <span class="keywordflow">if</span> (internal::isApprox(M(k, k), Scalar(0))) {</div>
<div class="line"><a name="l00162"></a><span class="lineno">  162</span>&#160;        <span class="keywordflow">return</span> <span class="keyword">false</span>;</div>
<div class="line"><a name="l00163"></a><span class="lineno">  163</span>&#160;      }</div>
<div class="line"><a name="l00164"></a><span class="lineno">  164</span>&#160; </div>
<div class="line"><a name="l00165"></a><span class="lineno">  165</span>&#160;      <span class="comment">// Make r orthogonal to q_i, i = 0..k-1</span></div>
<div class="line"><a name="l00166"></a><span class="lineno">  166</span>&#160;      Scalar beta = f(k) / M(k, k);</div>
<div class="line"><a name="l00167"></a><span class="lineno">  167</span>&#160;      r = r - beta * G.col(k);</div>
<div class="line"><a name="l00168"></a><span class="lineno">  168</span>&#160;      x = x + beta * U.col(k);</div>
<div class="line"><a name="l00169"></a><span class="lineno">  169</span>&#160;      normr = r.stableNorm();</div>
<div class="line"><a name="l00170"></a><span class="lineno">  170</span>&#160; </div>
<div class="line"><a name="l00171"></a><span class="lineno">  171</span>&#160;      <span class="keywordflow">if</span> (replacement &amp;&amp; normr &gt; tolb / mp) {</div>
<div class="line"><a name="l00172"></a><span class="lineno">  172</span>&#160;        trueres = <span class="keyword">true</span>;</div>
<div class="line"><a name="l00173"></a><span class="lineno">  173</span>&#160;      }</div>
<div class="line"><a name="l00174"></a><span class="lineno">  174</span>&#160; </div>
<div class="line"><a name="l00175"></a><span class="lineno">  175</span>&#160;      <span class="comment">// Smoothing:</span></div>
<div class="line"><a name="l00176"></a><span class="lineno">  176</span>&#160;      <span class="keywordflow">if</span> (smoothing) {</div>
<div class="line"><a name="l00177"></a><span class="lineno">  177</span>&#160;        t = r_s - r;</div>
<div class="line"><a name="l00178"></a><span class="lineno">  178</span>&#160;        <span class="comment">// gamma is a Scalar, but the conversion is not allowed</span></div>
<div class="line"><a name="l00179"></a><span class="lineno">  179</span>&#160;        Scalar gamma = t.dot(r_s) / t.stableNorm();</div>
<div class="line"><a name="l00180"></a><span class="lineno">  180</span>&#160;        r_s = r_s - gamma * t;</div>
<div class="line"><a name="l00181"></a><span class="lineno">  181</span>&#160;        x_s = x_s - gamma * (x_s - x);</div>
<div class="line"><a name="l00182"></a><span class="lineno">  182</span>&#160;        normr = r_s.stableNorm();</div>
<div class="line"><a name="l00183"></a><span class="lineno">  183</span>&#160;      }</div>
<div class="line"><a name="l00184"></a><span class="lineno">  184</span>&#160; </div>
<div class="line"><a name="l00185"></a><span class="lineno">  185</span>&#160;      <span class="keywordflow">if</span> (normr &lt; tolb || iter == maxit) {</div>
<div class="line"><a name="l00186"></a><span class="lineno">  186</span>&#160;        <span class="keywordflow">break</span>;</div>
<div class="line"><a name="l00187"></a><span class="lineno">  187</span>&#160;      }</div>
<div class="line"><a name="l00188"></a><span class="lineno">  188</span>&#160; </div>
<div class="line"><a name="l00189"></a><span class="lineno">  189</span>&#160;      <span class="comment">// New f = P&#39;*r (first k  components are zero)</span></div>
<div class="line"><a name="l00190"></a><span class="lineno">  190</span>&#160;      <span class="keywordflow">if</span> (k &lt; S - 1) {</div>
<div class="line"><a name="l00191"></a><span class="lineno">  191</span>&#160;        f.segment(k + 1, S - (k + 1)) = f.segment(k + 1, S - (k + 1)) - beta * M.block(k + 1, k, S - (k + 1), 1);</div>
<div class="line"><a name="l00192"></a><span class="lineno">  192</span>&#160;      }</div>
<div class="line"><a name="l00193"></a><span class="lineno">  193</span>&#160;    }  <span class="comment">// end for</span></div>
<div class="line"><a name="l00194"></a><span class="lineno">  194</span>&#160; </div>
<div class="line"><a name="l00195"></a><span class="lineno">  195</span>&#160;    <span class="keywordflow">if</span> (normr &lt; tolb || iter == maxit) {</div>
<div class="line"><a name="l00196"></a><span class="lineno">  196</span>&#160;      <span class="keywordflow">break</span>;</div>
<div class="line"><a name="l00197"></a><span class="lineno">  197</span>&#160;    }</div>
<div class="line"><a name="l00198"></a><span class="lineno">  198</span>&#160; </div>
<div class="line"><a name="l00199"></a><span class="lineno">  199</span>&#160;    <span class="comment">// Now we have sufficient vectors in G_j to compute residual in G_j+1</span></div>
<div class="line"><a name="l00200"></a><span class="lineno">  200</span>&#160;    <span class="comment">// Note: r is already perpendicular to P so v = r</span></div>
<div class="line"><a name="l00201"></a><span class="lineno">  201</span>&#160;    <span class="comment">// Preconditioning</span></div>
<div class="line"><a name="l00202"></a><span class="lineno">  202</span>&#160;    v = r;</div>
<div class="line"><a name="l00203"></a><span class="lineno">  203</span>&#160;    v = precond.solve(v);</div>
<div class="line"><a name="l00204"></a><span class="lineno">  204</span>&#160; </div>
<div class="line"><a name="l00205"></a><span class="lineno">  205</span>&#160;    <span class="comment">// Matrix-vector multiplication:</span></div>
<div class="line"><a name="l00206"></a><span class="lineno">  206</span>&#160;    t = A * v;</div>
<div class="line"><a name="l00207"></a><span class="lineno">  207</span>&#160; </div>
<div class="line"><a name="l00208"></a><span class="lineno">  208</span>&#160;    <span class="comment">// Computation of a new omega</span></div>
<div class="line"><a name="l00209"></a><span class="lineno">  209</span>&#160;    om = internal::omega(t, r, angle);</div>
<div class="line"><a name="l00210"></a><span class="lineno">  210</span>&#160; </div>
<div class="line"><a name="l00211"></a><span class="lineno">  211</span>&#160;    <span class="keywordflow">if</span> (om == RealScalar(0.0)) {</div>
<div class="line"><a name="l00212"></a><span class="lineno">  212</span>&#160;      <span class="keywordflow">return</span> <span class="keyword">false</span>;</div>
<div class="line"><a name="l00213"></a><span class="lineno">  213</span>&#160;    }</div>
<div class="line"><a name="l00214"></a><span class="lineno">  214</span>&#160; </div>
<div class="line"><a name="l00215"></a><span class="lineno">  215</span>&#160;    r = r - om * t;</div>
<div class="line"><a name="l00216"></a><span class="lineno">  216</span>&#160;    x = x + om * v;</div>
<div class="line"><a name="l00217"></a><span class="lineno">  217</span>&#160;    normr = r.stableNorm();</div>
<div class="line"><a name="l00218"></a><span class="lineno">  218</span>&#160; </div>
<div class="line"><a name="l00219"></a><span class="lineno">  219</span>&#160;    <span class="keywordflow">if</span> (replacement &amp;&amp; normr &gt; tolb / mp) {</div>
<div class="line"><a name="l00220"></a><span class="lineno">  220</span>&#160;      trueres = <span class="keyword">true</span>;</div>
<div class="line"><a name="l00221"></a><span class="lineno">  221</span>&#160;    }</div>
<div class="line"><a name="l00222"></a><span class="lineno">  222</span>&#160; </div>
<div class="line"><a name="l00223"></a><span class="lineno">  223</span>&#160;    <span class="comment">// Residual replacement?</span></div>
<div class="line"><a name="l00224"></a><span class="lineno">  224</span>&#160;    <span class="keywordflow">if</span> (trueres &amp;&amp; normr &lt; normb) {</div>
<div class="line"><a name="l00225"></a><span class="lineno">  225</span>&#160;      r = b - A * x;</div>
<div class="line"><a name="l00226"></a><span class="lineno">  226</span>&#160;      trueres = <span class="keyword">false</span>;</div>
<div class="line"><a name="l00227"></a><span class="lineno">  227</span>&#160;      replacements++;</div>
<div class="line"><a name="l00228"></a><span class="lineno">  228</span>&#160;    }</div>
<div class="line"><a name="l00229"></a><span class="lineno">  229</span>&#160; </div>
<div class="line"><a name="l00230"></a><span class="lineno">  230</span>&#160;    <span class="comment">// Smoothing:</span></div>
<div class="line"><a name="l00231"></a><span class="lineno">  231</span>&#160;    <span class="keywordflow">if</span> (smoothing) {</div>
<div class="line"><a name="l00232"></a><span class="lineno">  232</span>&#160;      t = r_s - r;</div>
<div class="line"><a name="l00233"></a><span class="lineno">  233</span>&#160;      Scalar gamma = t.dot(r_s) / t.stableNorm();</div>
<div class="line"><a name="l00234"></a><span class="lineno">  234</span>&#160;      r_s = r_s - gamma * t;</div>
<div class="line"><a name="l00235"></a><span class="lineno">  235</span>&#160;      x_s = x_s - gamma * (x_s - x);</div>
<div class="line"><a name="l00236"></a><span class="lineno">  236</span>&#160;      normr = r_s.stableNorm();</div>
<div class="line"><a name="l00237"></a><span class="lineno">  237</span>&#160;    }</div>
<div class="line"><a name="l00238"></a><span class="lineno">  238</span>&#160; </div>
<div class="line"><a name="l00239"></a><span class="lineno">  239</span>&#160;    iter++;</div>
<div class="line"><a name="l00240"></a><span class="lineno">  240</span>&#160; </div>
<div class="line"><a name="l00241"></a><span class="lineno">  241</span>&#160;  }  <span class="comment">// end while</span></div>
<div class="line"><a name="l00242"></a><span class="lineno">  242</span>&#160; </div>
<div class="line"><a name="l00243"></a><span class="lineno">  243</span>&#160;  <span class="keywordflow">if</span> (smoothing) {</div>
<div class="line"><a name="l00244"></a><span class="lineno">  244</span>&#160;    x = x_s;</div>
<div class="line"><a name="l00245"></a><span class="lineno">  245</span>&#160;  }</div>
<div class="line"><a name="l00246"></a><span class="lineno">  246</span>&#160;  relres = normr / normb;</div>
<div class="line"><a name="l00247"></a><span class="lineno">  247</span>&#160;  <span class="keywordflow">return</span> <span class="keyword">true</span>;</div>
<div class="line"><a name="l00248"></a><span class="lineno">  248</span>&#160;}</div>
<div class="line"><a name="l00249"></a><span class="lineno">  249</span>&#160; </div>
<div class="line"><a name="l00250"></a><span class="lineno">  250</span>&#160;}  <span class="comment">// namespace internal</span></div>
<div class="line"><a name="l00251"></a><span class="lineno">  251</span>&#160; </div>
<div class="line"><a name="l00252"></a><span class="lineno">  252</span>&#160;<span class="keyword">template</span> &lt;<span class="keyword">typename</span> MatrixType_, <span class="keyword">typename</span> Preconditioner_ = DiagonalPreconditioner&lt;<span class="keyword">typename</span> MatrixType_::Scalar&gt; &gt;</div>
<div class="line"><a name="l00253"></a><span class="lineno">  253</span>&#160;<span class="keyword">class </span>IDRS;</div>
<div class="line"><a name="l00254"></a><span class="lineno">  254</span>&#160; </div>
<div class="line"><a name="l00255"></a><span class="lineno">  255</span>&#160;<span class="keyword">namespace </span>internal {</div>
<div class="line"><a name="l00256"></a><span class="lineno">  256</span>&#160; </div>
<div class="line"><a name="l00257"></a><span class="lineno">  257</span>&#160;<span class="keyword">template</span> &lt;<span class="keyword">typename</span> MatrixType_, <span class="keyword">typename</span> Preconditioner_&gt;</div>
<div class="line"><a name="l00258"></a><span class="lineno">  258</span>&#160;<span class="keyword">struct </span>traits&lt;<a class="code" href="namespaceEigen.html">Eigen</a>::IDRS&lt;MatrixType_, Preconditioner_&gt; &gt; {</div>
<div class="line"><a name="l00259"></a><span class="lineno">  259</span>&#160;  <span class="keyword">typedef</span> MatrixType_ MatrixType;</div>
<div class="line"><a name="l00260"></a><span class="lineno">  260</span>&#160;  <span class="keyword">typedef</span> Preconditioner_ Preconditioner;</div>
<div class="line"><a name="l00261"></a><span class="lineno">  261</span>&#160;};</div>
<div class="line"><a name="l00262"></a><span class="lineno">  262</span>&#160; </div>
<div class="line"><a name="l00263"></a><span class="lineno">  263</span>&#160;}  <span class="comment">// namespace internal</span></div>
<div class="line"><a name="l00264"></a><span class="lineno">  264</span>&#160; </div>
<div class="line"><a name="l00306"></a><span class="lineno">  306</span>&#160;<span class="keyword">template</span> &lt;<span class="keyword">typename</span> MatrixType_, <span class="keyword">typename</span> Preconditioner_&gt;</div>
<div class="line"><a name="l00307"></a><span class="lineno"><a class="line" href="classEigen_1_1IDRS.html">  307</a></span>&#160;<span class="keyword">class </span><a class="code" href="classEigen_1_1IDRS.html">IDRS</a> : <span class="keyword">public</span> <a class="codeRef" href="../classEigen_1_1IterativeSolverBase.html">IterativeSolverBase</a>&lt;IDRS&lt;MatrixType_, Preconditioner_&gt; &gt; {</div>
<div class="line"><a name="l00308"></a><span class="lineno">  308</span>&#160; <span class="keyword">public</span>:</div>
<div class="line"><a name="l00309"></a><span class="lineno">  309</span>&#160;  <span class="keyword">typedef</span> MatrixType_ MatrixType;</div>
<div class="line"><a name="l00310"></a><span class="lineno">  310</span>&#160;  <span class="keyword">typedef</span> <span class="keyword">typename</span> MatrixType::Scalar Scalar;</div>
<div class="line"><a name="l00311"></a><span class="lineno">  311</span>&#160;  <span class="keyword">typedef</span> <span class="keyword">typename</span> MatrixType::RealScalar RealScalar;</div>
<div class="line"><a name="l00312"></a><span class="lineno">  312</span>&#160;  <span class="keyword">typedef</span> Preconditioner_ Preconditioner;</div>
<div class="line"><a name="l00313"></a><span class="lineno">  313</span>&#160; </div>
<div class="line"><a name="l00314"></a><span class="lineno">  314</span>&#160; <span class="keyword">private</span>:</div>
<div class="line"><a name="l00315"></a><span class="lineno">  315</span>&#160;  <span class="keyword">typedef</span> <a class="codeRef" href="../classEigen_1_1IterativeSolverBase.html">IterativeSolverBase&lt;IDRS&gt;</a> <a class="codeRef" href="../classEigen_1_1IterativeSolverBase.html">Base</a>;</div>
<div class="line"><a name="l00316"></a><span class="lineno">  316</span>&#160;  <span class="keyword">using</span> Base::m_error;</div>
<div class="line"><a name="l00317"></a><span class="lineno">  317</span>&#160;  <span class="keyword">using</span> Base::m_info;</div>
<div class="line"><a name="l00318"></a><span class="lineno">  318</span>&#160;  <span class="keyword">using</span> Base::m_isInitialized;</div>
<div class="line"><a name="l00319"></a><span class="lineno">  319</span>&#160;  <span class="keyword">using</span> Base::m_iterations;</div>
<div class="line"><a name="l00320"></a><span class="lineno">  320</span>&#160;  <span class="keyword">using</span> Base::matrix;</div>
<div class="line"><a name="l00321"></a><span class="lineno">  321</span>&#160;  <a class="codeRef" href="../namespaceEigen.html#a62e77e0933482dafde8fe197d9a2cfde">Index</a> m_S;</div>
<div class="line"><a name="l00322"></a><span class="lineno">  322</span>&#160;  <span class="keywordtype">bool</span> m_smoothing;</div>
<div class="line"><a name="l00323"></a><span class="lineno">  323</span>&#160;  RealScalar m_angle;</div>
<div class="line"><a name="l00324"></a><span class="lineno">  324</span>&#160;  <span class="keywordtype">bool</span> m_residual;</div>
<div class="line"><a name="l00325"></a><span class="lineno">  325</span>&#160; </div>
<div class="line"><a name="l00326"></a><span class="lineno">  326</span>&#160; <span class="keyword">public</span>:</div>
<div class="line"><a name="l00328"></a><span class="lineno"><a class="line" href="classEigen_1_1IDRS.html#a029d9925395d811e62939d5aacc3dbc3">  328</a></span>&#160;  <a class="code" href="classEigen_1_1IDRS.html#a029d9925395d811e62939d5aacc3dbc3">IDRS</a>() : m_S(4), m_smoothing(false), m_angle(RealScalar(0.7)), m_residual(false) {}</div>
<div class="line"><a name="l00329"></a><span class="lineno">  329</span>&#160; </div>
<div class="line"><a name="l00340"></a><span class="lineno">  340</span>&#160;  <span class="keyword">template</span> &lt;<span class="keyword">typename</span> MatrixDerived&gt;</div>
<div class="line"><a name="l00341"></a><span class="lineno"><a class="line" href="classEigen_1_1IDRS.html#a3407644df005d4a74ce7e8ca9f03c320">  341</a></span>&#160;  <span class="keyword">explicit</span> <a class="code" href="classEigen_1_1IDRS.html#a3407644df005d4a74ce7e8ca9f03c320">IDRS</a>(<span class="keyword">const</span> <a class="codeRef" href="../structEigen_1_1EigenBase.html">EigenBase&lt;MatrixDerived&gt;</a>&amp; A)</div>
<div class="line"><a name="l00342"></a><span class="lineno">  342</span>&#160;      : <a class="codeRef" href="../classEigen_1_1IterativeSolverBase.html">Base</a>(A.derived()), m_S(4), m_smoothing(false), m_angle(RealScalar(0.7)), m_residual(false) {}</div>
<div class="line"><a name="l00343"></a><span class="lineno">  343</span>&#160; </div>
<div class="line"><a name="l00349"></a><span class="lineno">  349</span>&#160;  <span class="keyword">template</span> &lt;<span class="keyword">typename</span> Rhs, <span class="keyword">typename</span> Dest&gt;</div>
<div class="line"><a name="l00350"></a><span class="lineno"><a class="line" href="classEigen_1_1IDRS.html#ab75c5ed081568173f7c32b18c1e7c013">  350</a></span>&#160;  <span class="keywordtype">void</span> <a class="code" href="classEigen_1_1IDRS.html#ab75c5ed081568173f7c32b18c1e7c013">_solve_vector_with_guess_impl</a>(<span class="keyword">const</span> Rhs&amp; b, Dest&amp; x)<span class="keyword"> const </span>{</div>
<div class="line"><a name="l00351"></a><span class="lineno">  351</span>&#160;    m_iterations = <a class="codeRef" href="../classEigen_1_1IterativeSolverBase.html#a168a74c8dceb6233b220031fdd756ba0">Base::maxIterations</a>();</div>
<div class="line"><a name="l00352"></a><span class="lineno">  352</span>&#160;    m_error = Base::m_tolerance;</div>
<div class="line"><a name="l00353"></a><span class="lineno">  353</span>&#160; </div>
<div class="line"><a name="l00354"></a><span class="lineno">  354</span>&#160;    <span class="keywordtype">bool</span> ret = internal::idrs(matrix(), b, x, Base::m_preconditioner, m_iterations, m_error, m_S, m_smoothing, m_angle,</div>
<div class="line"><a name="l00355"></a><span class="lineno">  355</span>&#160;                              m_residual);</div>
<div class="line"><a name="l00356"></a><span class="lineno">  356</span>&#160; </div>
<div class="line"><a name="l00357"></a><span class="lineno">  357</span>&#160;    m_info = (!ret) ? <a class="codeRef" href="../group__enums.html#gga85fad7b87587764e5cf6b513a9e0ee5ea1c6e20706575a629b27a105f07f1883b">NumericalIssue</a> : m_error &lt;= Base::m_tolerance ? <a class="codeRef" href="../group__enums.html#gga85fad7b87587764e5cf6b513a9e0ee5ea671a2aeb0f527802806a441d58a80fcf">Success</a> : <a class="codeRef" href="../group__enums.html#gga85fad7b87587764e5cf6b513a9e0ee5ea6a68dfb88a8336108a30588bdf356c57">NoConvergence</a>;</div>
<div class="line"><a name="l00358"></a><span class="lineno">  358</span>&#160;  }</div>
<div class="line"><a name="l00359"></a><span class="lineno">  359</span>&#160; </div>
<div class="line"><a name="l00361"></a><span class="lineno"><a class="line" href="classEigen_1_1IDRS.html#aeb11b800a0216ebf60560ec7f7e0e76a">  361</a></span>&#160;  <span class="keywordtype">void</span> <a class="code" href="classEigen_1_1IDRS.html#aeb11b800a0216ebf60560ec7f7e0e76a">setS</a>(<a class="codeRef" href="../namespaceEigen.html#a62e77e0933482dafde8fe197d9a2cfde">Index</a> S) {</div>
<div class="line"><a name="l00362"></a><span class="lineno">  362</span>&#160;    <span class="keywordflow">if</span> (S &lt; 1) {</div>
<div class="line"><a name="l00363"></a><span class="lineno">  363</span>&#160;      S = 4;</div>
<div class="line"><a name="l00364"></a><span class="lineno">  364</span>&#160;    }</div>
<div class="line"><a name="l00365"></a><span class="lineno">  365</span>&#160; </div>
<div class="line"><a name="l00366"></a><span class="lineno">  366</span>&#160;    m_S = S;</div>
<div class="line"><a name="l00367"></a><span class="lineno">  367</span>&#160;  }</div>
<div class="line"><a name="l00368"></a><span class="lineno">  368</span>&#160; </div>
<div class="line"><a name="l00375"></a><span class="lineno"><a class="line" href="classEigen_1_1IDRS.html#ad6f2fc19379d881e15d8c63f6ca8951a">  375</a></span>&#160;  <span class="keywordtype">void</span> <a class="code" href="classEigen_1_1IDRS.html#ad6f2fc19379d881e15d8c63f6ca8951a">setSmoothing</a>(<span class="keywordtype">bool</span> smoothing) { m_smoothing = smoothing; }</div>
<div class="line"><a name="l00376"></a><span class="lineno">  376</span>&#160; </div>
<div class="line"><a name="l00387"></a><span class="lineno"><a class="line" href="classEigen_1_1IDRS.html#afb154a59c37ef4411391cb3b19aed4c9">  387</a></span>&#160;  <span class="keywordtype">void</span> <a class="code" href="classEigen_1_1IDRS.html#afb154a59c37ef4411391cb3b19aed4c9">setAngle</a>(RealScalar angle) { m_angle = angle; }</div>
<div class="line"><a name="l00388"></a><span class="lineno">  388</span>&#160; </div>
<div class="line"><a name="l00392"></a><span class="lineno"><a class="line" href="classEigen_1_1IDRS.html#a002e59270919dcb1ad6f70de3e900dcf">  392</a></span>&#160;  <span class="keywordtype">void</span> <a class="code" href="classEigen_1_1IDRS.html#a002e59270919dcb1ad6f70de3e900dcf">setResidualUpdate</a>(<span class="keywordtype">bool</span> update) { m_residual = update; }</div>
<div class="line"><a name="l00393"></a><span class="lineno">  393</span>&#160;};</div>
<div class="line"><a name="l00394"></a><span class="lineno">  394</span>&#160; </div>
<div class="line"><a name="l00395"></a><span class="lineno">  395</span>&#160;}  <span class="comment">// namespace Eigen</span></div>
<div class="line"><a name="l00396"></a><span class="lineno">  396</span>&#160; </div>
<div class="line"><a name="l00397"></a><span class="lineno">  397</span>&#160;<span class="preprocessor">#endif </span><span class="comment">/* EIGEN_IDRS_H */</span><span class="preprocessor"></span></div>
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<div class="ttc" id="aclassEigen_1_1IDRS_html_a3407644df005d4a74ce7e8ca9f03c320"><div class="ttname"><a href="classEigen_1_1IDRS.html#a3407644df005d4a74ce7e8ca9f03c320">Eigen::IDRS::IDRS</a></div><div class="ttdeci">IDRS(const EigenBase&lt; MatrixDerived &gt; &amp;A)</div><div class="ttdef"><b>Definition:</b> IDRS.h:341</div></div>
<div class="ttc" id="aclassEigen_1_1IDRS_html_ab75c5ed081568173f7c32b18c1e7c013"><div class="ttname"><a href="classEigen_1_1IDRS.html#ab75c5ed081568173f7c32b18c1e7c013">Eigen::IDRS::_solve_vector_with_guess_impl</a></div><div class="ttdeci">void _solve_vector_with_guess_impl(const Rhs &amp;b, Dest &amp;x) const</div><div class="ttdef"><b>Definition:</b> IDRS.h:350</div></div>
<div class="ttc" id="aclassEigen_1_1IDRS_html_ad6f2fc19379d881e15d8c63f6ca8951a"><div class="ttname"><a href="classEigen_1_1IDRS.html#ad6f2fc19379d881e15d8c63f6ca8951a">Eigen::IDRS::setSmoothing</a></div><div class="ttdeci">void setSmoothing(bool smoothing)</div><div class="ttdef"><b>Definition:</b> IDRS.h:375</div></div>
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